The salient point here is that we arrive at a result similar to that in [1]} without the assumption of ergodicity or positivity of the Lyapunov exponent; we just need positive upper Lyapunov exponent. It is possible to view an ergodic family of operators as similar to a family of operators with a phase boundary condition. This surface analogy would lead us to believe that the result in [1]} should be the same as the result in our situation, but this is not the case. Moreover, if there were an analogy between the ergodic parameter and the phase boundary parameter, then Simon's result would lead one to believe that a Lebesgue null set of phases needs to be excluded; our result shows that this is not true. We are able to obtain a logarithmic bound for all boundary phases.
